鲁棒VaR与CVaR优化：联合分布、均值与协方差模糊性
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本文主要探讨了在金融风险管理中如何在分布、均值和协方差的联合不确定性下进行鲁棒的VaR（Value at Risk）和CVaR（Conditional Value at Risk）优化。
文章详细介绍了在风险管理和投资组合优化领域的最新研究进展。作者Somayyeh Lotfi和Stavros A. Zenios首先提出了一个关键问题：在面临数据模糊性，即不确定性的环境中，如何确保投资策略的稳健性。他们关注的是VaR和CVaR这两种广泛使用的风险度量方法，这两种方法在传统的金融理论中对于预测潜在损失具有重要意义。
VaR是衡量金融市场中资产组合可能遭受的最大损失，而CVaR则进一步考虑了超出VaR的概率分布，提供了更全面的风险评估。然而，在实际应用中，由于市场数据的不完整性和复杂性，分布、均值和协方差往往存在不确定性，这使得基于这些参数的风险度量可能变得不可靠。
为了解决这一问题，作者提出了一种新的鲁棒优化模型。他们考虑了椭球形、多面体和区间不确定性集合（即ellipsoidal, polytopic, 和interval ambiguity sets）来刻画分布、均值和协方差的不确定性。这些模型旨在确保即使在数据有误或未来市场条件变化的情况下，投资策略也能保持稳定。
在论文中，作者详细阐述了如何构建这些模型，并分析了每种模型的特性。他们讨论了不同不确定性集合理论基础，以及这些模型在处理Eurozone危机等实际经济事件时的表现。此外，通过数值实验，作者展示了这些鲁棒优化模型在应对数据模糊性方面的优势，以及它们如何帮助投资者制定更为稳健的投资决策。
这篇论文对金融风险管理人员和投资策略制定者具有重要的指导价值，它提供了一套工具来处理风险度量中的不确定性，从而提高投资组合的稳健性。这种方法不仅适用于学术研究，而且可以应用于实际的金融产品设计和风险管理实践中。
560 S. Lotfi, S.A. Zenios / European Journal of Operational Research 269 (2018) 556–576
Proposition 1. If random variable ξ has a distribution from the set
D with fixed ¯μ and
¯
, then
min
γ ∈ R
max
π∈ D
F
α
(x, γ ) = min
γ ∈ R
γ (12)
s.t. max
π∈ D
Prob { γ ≤ f (x, ξ ) } ≤ 1 − α,
and the solution is −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 −α
x
¯
x .
Proof. From Eq. (10) in Pac and Pınar (2014) we know that
min
γ ∈ R
max
π∈ D
F
α
(x, γ ) = −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x . (13)
For the constraint of the optimization problem in the right-hand
side of (12) , we use Theorem 1 of Ghaoui et al. (2003) (set 1 − α
and f ( x , ξ ) instead of and −r(w, x ) , respectively), which means
max
π∈ D
Prob { γ ≤ f (x, ξ ) } ≤ 1 − α
is equivalent to
−r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x ≤ γ .
Hence, the optimization problem in the right-hand side is equiva-
lent to
min
γ ∈ R
γ (14)
s.t. −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 − α
x
¯
x ≤ γ ,
which has the minimum value −r
f
− ( ¯μ − r
f
e )
x +
√
α
√
1 −α
x
¯
x .
This completes the proof.
Remark
6. The left and right-hand sides of (12) are RCVaR
and RVaR, respectively, associated with the ambiguity set of
Definition 1 . Hence, the robust counterpart to VaR and CVaR op-
timization for distribution ambiguity is the same optimization
model.
Zymler, Kuhn, and Rustem (2013a,b) also derived worst-case
VaR and CVaR for piece-wise and quadratic loss functions when
there is ambiguity in distribution of asset returns but known mean
and covariance information. Both of optimization problems are ob-
tained as SDPs and they turned out to be identical for both loss
functions. When the loss function is linear (no derivative securi-
ties), the associated problems reduce to the same SOCPs as in our
Proposition 1 . Zymler et al. (2013a) extend further these identical
models to ambiguity in means and covariance matrix. They take
into account box-type ambiguity in the moment matrix but we
consider joint ellipsoidal ambiguity set for mean and covariance
matrix and obtain the robust models for a linear loss function next.
So, compared to Zymler et al. we establish the equivalence with
more general ambiguity sets when there is ambiguity in means
and covariance matrix, whereas they use more general loss func-
tions. Our result also generalizes the result of
ˇ
Cerbáková (2006)
for
the special case of symmetric distributions identified only by the
first two moments. It also establishes that the bounds obtained by
Bertsimas, Lauprete, and Samarov (2004) on VaR and CVaR for dis-
tribution ambiguity are tight.
We obtain now RVaR and RCVaR optimization models for am-
biguity in distributions, mean returns and covariance matrix.
Theorem 2. If random variable ξ has a distribution from the set D
and ( ¯μ,
¯
) ∈ U
δ
( ˆ μ,
ˆ
) . Then, the robust counterpart to VaR portfo-
lio optimization model (5) and the robust counterpart to CVaR model
(6) are both represented by the following SOCP:
min
x ∈ R
n
− r
f
− ( ˆ μ − r
f
e )
x +
max
κ∈ [0 , 1]
f (κ )
ˆ
1
2
x (15)
s.t. −
δ
√
S
ˆ
1
2
x + ( ˆ μ − r
f
e )
x ≥ d − r
f
,
where f (κ ) =
√
α
√
1 −α
(1 + δ
2(1 −κ )
S−1
+ δ
κ
S
.
Proof. See Appendix A.1 .
Remark
7. f ( κ) is a strictly concave function with lim
κ→ 0
f
(κ ) =
∞ and lim
κ→ 1
f
(κ ) = −∞ . Hence, f ( κ) has a unique maximum in
the interval (0, 1).
3.2. Constructing the ambiguity set
The ambiguity set is typically taken as input in the robust op-
timization literature. In some cases the ambiguity set can be de-
fined as the confidence regions of the statistical estimators of the
model parameters ( Goldfarb & Iyengar, 2003; Schottle & Werner,
2009 ). For other cases –such as in the applications we solve later—
we
may be given multiple estimates of model parameters which
raises the question as to what is the appropriate ambiguity set.
In finance, for instance, it is not uncommon to be given estimates
by multiple securities analysts and we need a method to construct
an ambiguity set, including its center. This issue has not been ad-
dressed in existing literature and we now propose and solve ana-
lytically a nonlinear SDP for finding the center of a joint ellipsoidal
set.
Assume K experts provide estimates for mean returns and co-
variance matrices ( ¯μ
k
,
¯
k
) , k = 1 , 2 , . . . , K. (For convenience we as-
sume they were all estimated using the same number of scenarios
S .) To construct their joint ellipsoidal ambiguity set we need to fix
the center ( ˆ μ,
ˆ
). This is obtained as the solution of a nonlinear
convex program for minimizing the l
2
-norm of the parameters δ
k
,
for k = 1 , 2 , . . . , K, where each parameter corresponds to the ellip-
soid with center ( ˆ μ,
ˆ
) containing observation ( ¯μ
k
,
¯
k
) . Referring
to Definition 2 the optimization problem is given by
min
ˆ μ∈ R
n
,
ˆ
∈ S
n
++
K
k =1
S ( ¯μ
k
− ˆ μ)
ˆ
−1
( ¯μ
k
− ˆ μ) +
S − 1
2
ˆ
−
1
2
(
¯
k
−
ˆ
)
ˆ
−
1
2
2
tr
,
(16)
where S
n
++
is the set of all n -dimensional, symmetric, positive def-
inite matrices. This problem is equivalent to
min
ˆ μ∈ R
n
,
ˆ
∈ S
n
++
K
k =1
S ( ¯μ
k
− ˆ μ)
ˆ
−1
( ¯μ
k
− ˆ μ) +
S − 1
2
ˆ
−
1
2
(
¯
k
−
ˆ
)
ˆ
−
1
2
2
tr
.
(17)
The next theorem gives the solution of this problem, if a solu-
tion exists.
Theorem 3. If (17) is solvable, then it admits the following solution:
1. ˆ μ =
1
K
K
k =1
¯μ.
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